Vector optimization

Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to certain constraints. A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.

Problem formulation

In mathematical terms, a vector optimization problem can be written as:

$C\operatorname{-}\min_{x \in S} f(x)$

where $f: X \to Z$ for a partially ordered vector space $Z$. The partial ordering is induced by a cone $C \subseteq Z$. $X$ is an arbitrary set and $S \subseteq X$ is called the feasible set.

Solution concepts

There are different minimality notions, among them:

• $\bar{x} \in S$ is a weakly efficient point (weak minimizer) if for every $x \in S$ one has $f(x) - f(\bar{x}) \not\in -\operatorname{int} C$.
• $\bar{x} \in S$ is an efficient point (minimizer) if for every $x \in S$ one has $f(x) - f(\bar{x}) \not\in -C \backslash \{0\}$.
• $\bar{x} \in S$ is a properly efficient point (proper minimizer) if $\bar{x}$ is a weakly efficient point with respect to a closed pointed convex cone $\tilde{C}$ where $C \backslash \{0\} \subseteq \operatorname{int} \tilde{C}$.

Every proper minimizer is a minimizer. And every minimizer is a weak minimizer.^[1]

Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.^[2]

Solution methods

• Benson's algorithm for linear vector optimization problems^[2]

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

$\mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x)$

where $f: X \to \mathbb{R}^d$ and $\mathbb{R}^d_+$ is the non-negative orthant of $\mathbb{R}^d$. Thus the minimizer of this vector optimization problem are the Pareto efficient points.

References

1. Ginchev, I.; Guerraggio, A.; Rocca, M. (2006). "From Scalar to Vector Optimization". Applications of Mathematics 51: 5. doi:10.1007/s10492-006-0002-1.
2. Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.